Multistep correction for angle consistent artifacts in radial sampled datasets

ABSTRACT

This document discloses, among other things, a method for removing a bullseye artifact from a radial image generated using magnetic resonance and using a swept frequency pulse.

CLAIM OF PRIORITY

This application claims the benefit of priority to U.S. patentapplication Ser. No. 61/505,672, filed Jul. 8, 2012, the entiredisclosure of which is hereby incorporated by reference herein in itsentirety.

CROSS-REFERENCE TO RELATED PATENT DOCUMENTS

This patent application is related to Steen Moeller, U.S. ProvisionalPatent Application Ser. No. 61/043,984, entitled “RF PULSE DISTORTIONCORRECTION,” filed on Apr. 10, 2008, and related to Steen Moeller, U.S.patent application Ser. No. 12/384,888, entitled “RF PULSE DISTORTIONCORRECTION,” filed on Apr. 9, 2009, each of which are incorporated byreference herein in their entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with Government support under 5P41RR008079 awardby the National Institutes of Health. The Government has certain rightsin this invention.

BACKGROUND

An image acquired with radially distributed projections can sometimesinclude undesirable artifacts. A variety of factors can cause artifacts.Some artifacts can be ignored and yet other artifacts can render animage useless or lead to a misdiagnosis.

One example of an image artifact resembles a bullseye having generallyconcentric rings of alternating dark and light shading.

OVERVIEW

The present subject matter includes a method for reducing or correctingan artifact due to angularly consistent errors in 2d or 3d radialsampled datasets. In one example the angularly consistent error may beintroduced by a previous artifact correction routine.

Examples of the present subject matter can be applied to radial magneticresonance imaging, radial sampled imaging systems (detector error) inSPECT, PET, CT and other angularly sampled imaging systems.

The present subject matter includes a system and a method for correctinga bullseye artifact. A bullseye artifact may appear in a radial imagegenerated, for example, using the magnetic resonance protocol referredto as SWIFT. The SWIFT protocol, derived from swept imaging with Fouriertransform, uses a frequency modulated pulse to accomplish nearlysimultaneous excitation and reception.

In one example, a bullseye artifact of the image can be partiallycorrected using a first procedure and a remaining portion of thebullseye artifact can be corrected using a second procedure. Either orboth of the first procedure and the second procedure can include acalculation performed in the time domain or using a calculationperformed in the frequency domain.

One example includes a method that includes calculating a first averageprojection and calculating a first bullseye correction. The firstbullseye correction is then applied to the dataset and thereafter, theprocedure continues with calculating a second average projection andcalculating a second bullseye correction. In addition, the methodincludes applying an inverse bullseye correction based on the secondbullseye correction.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings, which are not necessarily drawn to scale, like numeralsmay describe similar components in different views. Like numerals havingdifferent letter suffixes may represent different instances of similarcomponents. The drawings illustrate generally, by way of example, butnot by way of limitation, various embodiments discussed in the presentdocument.

FIG. 1 includes a series of images that exhibit a bullseye artifact.

FIG. 2 illustrates bullseye artifact correction using time domaincorrelation.

FIG. 3 includes a block diagram of an imaging system.

FIGS. 4A and 4B include flow charts for adjusting an image.

FIG. 5 illustrates a SWIFT flow diagram.

FIG. 6 illustrates a timewise view of SWIFT.

FIGS. 7A and 7B illustrate simulated FID data having no gaps in receiveddata.

FIGS. 8A and 8B illustrate simulated FID data having transmitter gaps.

FIGS. 9A and 9B illustrate simulated FID data having both transmittergaps and receiver gaps.

FIGS. 10A and 10B illustrate simulated data in the frequency spectrum.

FIG. 11 illustrates an image of a phantom after bullseye correction.

FIGS. 12A and 12B illustrate images of a phantom after inverse bullseyecorrection.

FIGS. 13A and 13B illustrate subtraction images.

FIG. 14 illustrates a timewise plot of bullseye correction and inversebullseye correction.

FIGS. 15A, 15B, and 15C include sinograms.

FIG. 16 illustrates a relationship between excitation bandwidth and flipangle.

FIG. 17 illustrates data corresponding to simulated flip angles.

FIGS. 18A-18J graphically illustrates selected components of particularequations.

FIGS. 19A-19C illustrate a SWIFT pulse sequence.

FIGS. 20A and 20B illustrate profiles of a baseband.

DETAILED DESCRIPTION

Certain excitation signals may lead to distortion in the received signalor generated image. For example, an image generated using a frequencyswept pulse may include objectionable distortion appearing as a bullseyeartifact.

FIG. 1 includes a series of whole head images that exhibit a bullseyeartifact. The whole head images illustrate good resolution and contrast.The three images in the figure are generated using MRI at 4 T and wereacquired using SWIFT with a hyperbolic secant (HS1) pulse, TR=6.1 ms,nominal flip angle=4°, 96000 independent spokes, FOV=35×35×35 cm,acquisition time≈10 min, isotropic resolution of 1.37×1.37×1.37. Thebullseye artifact appears as generally concentric rings of alternatingdark and light regions.

Radial sampling depicts a projection that can be thought of as lookingat a hub along a particular spoke of a wheel. The magnetic resonancescanner provides data representing the amplitude component and the phasecomponent of an image. A Fourier transform can be used to shift betweenthe time domain and the frequency domain. In radial imaging, griddingand an inverse Fourier transform of the k-space data yields an image. AFourier transform of a projection yields a particular ‘spoke’ in radialk-space.

With the SWIFT protocol, for example, the excitation signal actuallyapplied to a region of interest may differ from that which was intended.The excitation signal can be modeled using the intended signal and anerror component. The error component can arise from non-linearities inthe amplifier as well as other factors. In the present subject matter,the error is assumed to be temporally consistent and the image can becorrected by application of a suitable algorithm.

The bullseye artifact can be viewed as high frequency modulation overthe bandwidth of the response. Accordingly, selection of a suitablefilter for the system response can remove the high frequency componentand provide a clean image. The filter can be selected to provide asmooth average projection.

In one example, the calculated data is multiplied by a low pass filteredaverage signal and divided by an unfiltered average signal.

In addition to SWIFT, other magnetic resonance sequences can be used forradial sampling and are thus correctable by the present subject matter.The present subject matter can be used with any magnetic resonancesequence that uses sequential excitement of different frequencies.

The 3D radial MRI technique known as SWIFT exploits a frequencymodulated pulse to accomplish nearly simultaneous excitation andreception. SWIFT uses the concept that the resulting spectrum isexpressible as the convolution of the resulting NMR signal with theRF-pulse that the spins experience. In this approach, removal of the RFpulse from the NMR signal through correlation (a.k.a. deconvolution)requires accurate knowledge of the RF waveform that the spinsexperience. Yet, due to distortions occurring in the transmitterchannel, this function is typically different from the digital waveformthat was requested. There are several sources for consistent distortionsof the pulse shape including amplifier non-linearity and amplitudeinduced phase modulation, which can be compensated by amplifier mapping,and smaller phase delays in the resonant circuitry, which requires morechallenging hardware solutions.

With SWIFT, persistent RF distortion is observable as a bullseyeartifact, which can degrade image quality. The present subject matterincludes a method for correcting these deleterious effects, withoutexplicit knowledge of them or their origin.

TABLE 1 r(t) = h(t) 

 x(t) 

 R(ω) = H(ω)X(ω) (1) H_(calc) = R/X_(exp) (2) H_(calc) = (X/X_(exp))H(3) H_(calc) = Σ_(all views)H_(calc) & H_(calc) = LP (H_(calc)) ≈ H (4)$H_{corrected} = {\frac{\overset{\_}{\overset{\_}{H_{calc}}}H_{calc}}{\overset{\_}{H_{calc}}} = {\frac{{\overset{\_}{\overset{\_}{H_{calc}}}\left( {X/X_{\exp}} \right)}H}{\overset{\_}{H}\left( {X/X_{\exp}} \right)} \approx H}}$(5)

The measured signal r(t) can be expressed as in Equation 1 of Table 1,where x(t) is the RF pulse that the spins experience, and h(t) is thefree induction decay (FID) due to short pulse excitation (impulseresponse).

The correlation with an expected pulse function x_(exp)(t) in thefrequency domain gives a spectrum H_(calc) (Equation 2), which isdifferent from the spectrum of the spin system H (Equation 3). Here, theRF pulse distortions are considered as any persistent effect, i.e.,x(t)=x_(exp)(t)+Δx(t). For radial acquisitions, the persistentdistortion of the RF pulse can be estimated from the data itself. Thepremise for doing this is that the average projection of all directionsis smooth, and that the error is the same for all views.

The corrected projection can then be determined from Equation 5, whichentails division with the average projection H_(calc) and multiplicationwith the filtered average projection H_(calc) .

A correction function can be determined as shown in Equation 1 of Table1 and implemented in either the frequency domain or in the time domain.The correction function is defined as:

${{{correction}\mspace{14mu}{function}} \equiv H_{crf}} = \frac{{\overset{\_}{\overset{\_}{H}}}_{calc}}{{\overset{\_}{H}}_{calc}}$

The correction function can be applied in either the frequency domainusing H_(corrected)=H_(crf)H_(calc) or the correction function can beapplied in the time domain as {hacek over ( )}H_(corrected)={hacek over( )}H_(crf){circle around (x)}{hacek over ( )}H_(calc) where {hacek over( )}H_(corrected) is the Fourier transform of H_(corrected). Thecorrection function can be expressed as a ratio and, in one example, isdetermined by performing a spline interpolation.

FIG. 2 illustrates bullseye artifact correction using time domaincorrelation. In the figure, the abscissa shows the time domain aftercorrelation and the ordinate denotes the logarithm of the signalamplitude. The line marked with * (asterisk) symbol illustrates the freeinduction decay (FID) after correlation and before bullseye artifactcorrection. In this line, the high spatial frequencies are attenuated.The line marked with □ (square shape) symbol is the FID after bullseyeartifact correction in the frequency domain and the Fourier transformedto the time domain. In this line, the attenuation of the highfrequencies in the FID is removed. The line The line with • (dot) symbolis the FID in the time domain after correlation with the bullseyecorrection function in the time domain. The bullseye correction functionin the time domain is the Fourier transform of the “correction function”described above. In this line, the attenuation of the high frequenciesin the FID is removed. The figure illustrates that the parallelismbetween the time domain and the frequency domain is imperfect for thosesignals having a finite duration.

Measurements from both 9.4 T animal and 4 T human systems, on whichSWIFT is implemented, have been investigated for exhibiting thesebullseye artifacts. In one example, the function H_(calc) , whichrepresents the low-pass filtering of H_(calc) was estimated with astrongly windowed FFT (a hamming window to the power 30). This wasconsistently found to remove the artifact. Corrections with standardhamming filtering and subsequent polynomial filtering (Savitzky-Golay,5^(th) order and frame size 61) can also be used. The images of FIG. 1were constructed using gridding with a Kaiser-Bessel function. Griddingwith a Kaiser-Bessel function includes image reconstruction byinterpolating the acquired data onto a Cartesian grid using aKaiser-Bessel convolution kernel followed by fast inverse Fouriertransform.

The function H_(calc) can be estimated with a FFT (fast Fouriertransform) not described as “strongly windowed.” Windowing refers to atechnique used to shape the time portion of measurement data in order toreduce edge effects that result in spectral leakage in the FFT spectrum.A variety of window functions can be used with aperiodic data, such asthat which may be encountered using the present subject matter. Criteriawith which to select a particular window function can include, by way ofexamples, frequency resolution, spectral leakage, and amplitudeaccuracy.

Pulse imperfections due to electronic components and fundamentalphysical limitations are real and confounding factors for performingadvanced MR experiments. Yet, with the future of MR tending towardnon-Cartesian radial sequences and frequency swept pulses, correctionsfor such effects are needed. According to one theory, the dominantbullseye effect is caused by spins experiencing a distorted RF waveform.The present subject matter includes a data driven method for correction,thus leading to images of clinical quality.

Bullseye removal, according to the present subject matter, entails noadditional acquisitions or modifications to existing protocols. Longrange (low frequency) perturbations relative to the object can berealized as an intensity modulation, and can be correctable by intensitycorrection techniques—which cannot handle high frequency modulations.

In addition to the post-processing technique, additional directmeasurements of the RF pulse have shown that the solution of Equation 1can be corrected by correlation using the measured RF.

In one example, the present subject matter is implemented using a filterhaving a particular pass band. The filter, as noted elsewhere in thisdocument, can include a low pass filter in which a low frequencyspectral component is passed with little or no attenuation and a highfrequency spectral component is attenuated. In one example, the filteris configured to attenuate or remove the high (magnitude) spatialfrequency components, (i.e. high positive and negative frequencies). Thefilter can include a Gaussian filter, a polynomial filter, or other typeof filter.

In SWIFT, useful signal is extracted by correlation of the spin systemresponse with the pulse function. Misrepresentation of the pulsefunction results in a systematic error in the resulting spectrum. Such asystematic error in radial imaging shows up in the reconstructed imageas a bullseye artifact. To avoid such artifacts, the pulse function usedfor correlation must be as faithful as possible to the physical pulsetransmitted by the RF coil. As shown herein, the bullseye artifact canbe corrected by application of a correction function.

In one example, the pulse function applied is determined by a coilconfigured to detect an RF signal. The coil, sometimes referred to as asniffer coil, can include a probe having a wound electrical element withsensitivity to the frequencies of interest. In one example, a sample ofthe signal is generated by a sample port coupled to the magneticresonance system.

The present subject matter can be used with data corresponding tovarious types of imaging modalities and using various ways ofcalculating the multiplicative ratio factor used to remove the bullseyeartifact. In one particular example, the present subject matter is usedwith magnetic resonance data. Other types of image data, besidesmagnetic resonance, can also be used with the present subject matter.

In addition, the ratio can be obtained using alternative methods. In oneexample, the ratio is generated using an average projection and afiltered average projection based on the data.

The excitation signal can be frequency modulated or it can be amplitudemodulated. In one example, the excitation signal includes an adiabaticpulse.

In addition to using the received data, various methods can be used todetermine the filtered average projection.

An example of the present subject matter includes both bullseyecorrection and inverse bullseye correction. A corrected image can begenerated by applying bullseye correction to image data, as described inthis document, followed by an inverse bullseye correction. The inversebullseye correction method can include multiplication by a weightingfactor. In one example, the weighting factor is ½.

An example of the present subject matter includes calculating a firstaverage projection and calculating a first bullseye correction. Thefirst bullseye correction is applied to the image data to produce abullseye corrected image data. In addition, a second average projectionis calculated based on the corrected image data and a second bullseyecorrection is calculated based on the corrected image data. An inverseof the second bullseye correction is applied to the corrected image datato produce inverse bullseye corrected image data. The inverse bullseyecorrected image data can remove a substantial portion of objectdependent artifacts.

FIG. 3 includes a block diagram of imaging system 300. Imaging system300, in this example, depicts a magnetic resonance system and includesmagnet 305. Receiver coil 315 and object 310 are positioned within thefield of magnet 305. Object 310 can include a human body, an animal, aphantom, or other specimen. Receiver coil 315, sometimes referred to asan antenna, provides a signal to receiving unit 320. Receiving unit 320can include an analog-to-digital converter (ADC), a pre-amplifier, afilter, or other module to prepare the received signal for processing.Receiving unit 320 is coupled to processor 330. In one example, system300 includes a transmit unit and a transmit coil configured to provideexcitation to object 310.

Processor 330 can include a digital signal processor, a microprocessor,a controller, or other module to perform an operation on the signal fromthe receiving unit. Processor 330 is also coupled to storage 325,display 335 and output unit 340.

Storage 325 can include a memory for storing data. The data can includeimage data as well as results of processing performed by processor 330.In one example, storage 325 provides storage for executable instructionsfor use by processor 330. The instructions can be configured toimplement a particular algorithm.

Display 335 can include a screen, a monitor, or other device to render avisible image corresponding to object 310. For example, display 335 canbe configured to display a radial projection, a Cartesian coordinateprojection, or other view corresponding to object 310. Output unit 340can include a printer, a storage device, a network interface or otherdevice configured to receive processed data.

FIG. 4A includes a flow chart of method 400 for adjusting an image.Method 400 represents an example of processing configured to correct abullseye artifact in an image.

At 405, radial image data is received. The radial image can be derivedfrom MR data or data derived using other image modalities such asultrasonic, PET, or CT. The radial image data, in one example, isderived from an MRI system (such as system 300 of FIG. 3) configured toimplement a SWIFT protocol of excitation and readout.

In the example shown, at 410, the image data is pre-compensated. Signalintensity varies from view to view, and pre-compensation can be appliedto H_(calc)(ƒ) in order to reduce the amount of variation. In variousexample, the compensation function is either 1/t or 1/t², and is appliedin the time domain, as precompensated H_(calc)=^(({hacek over ()}H_(calc))/t) or precompensated H_(calc)=^(({hacek over ()}H_(calc))/t²) where {hacek over ( )}H_(calc) is the Fourier transformand ^H_(calc) is the inverse Fourier transform.

At 415, the correction factor is determined. In one example, thisincludes determining a ratio of an average projection and a filteredaverage projection.

At 420, bullseye correction is performed on the pre-compensated data.After correction for RF imperfections (such as bullseye artifact), thedata is post-compensated with the inverse correction factor (e.g. t ort²) at 425. Post-compensation entails recovery of the data for thecorrected image.

In the example shown, at 430, the corrected image is generated using thecompensated and bullseye corrected data.

Compensation (including pre-compensation and post-compensation) can beeffective for removing outliers from the data. In addition, compensationcan be applied in the frequency domain when the bullseye correction isperformed in the time domain.

Variations in the method are also contemplated. For example,pre-compensation (at 410) and post-compensation (at 425) can be omittedin bullseye artifact correction. As such, the range of data remainsun-compensated.

Method 400 can be effective for removing a first portion of a bullseyeartifact. Other portions of a bullseye artifact can be removed using anadditional method or by using multiple iterations of method 400 inconjunction with an additional method.

FIG. 4B includes an example of method 5 configured for removing aportion of a bullseye artifact. At 10, method 5 includes calculating anaverage projection and calculating a bullseye correction. At 20, method5 includes applying the bullseye correction to the data.

In one example, method 400 (FIG. 4A) is configured to provide a resultcomparable to a combination of 10 and 20 of method 5 (FIG. 4B).

As shown in FIG. 4B, at 30, method 5 includes again calculating anaverage projection and calculating a bullseye correction. At 40, method5 includes applying an inverse of the bullseye correction to the data.The application of an inverse of the bullseye correction is effectivefor removing some of the object dependent artifacts.

Variations of the above procedure are also contemplated. Followingapplication of inverse bullseye correction at 40, one example of method5 includes a feedback path 65 that returns to calculating an averageprojection and calculating a bullseye correction, at 10. Another exampleof method 5 includes, following calculating an average projection andcalculating a bullseye correction, at 30, feedback path 55 that returnsto calculating an average projection and calculating a bullseyecorrection, at 10.

Bullseye correction and inverse bullseye correction can be iterativelyexecuted to remove artifacts. Any number of iterations or repeatedcalculations can be used.

As described elsewhere in this document, one implantation of SWIFTincludes a plurality of gaps in the transmitted RF pulse. The gaps areused to insert intervals of single or multiple analog-to-digital (A/D)conversions to receive the NMR/MRI signal. The rapid transmit andreceive gating (producing short gaps in the pulse) introduces artifactsinto the transmitted pulse (data, and indirectly, into the receiveddata). FIGS. 5 and 6 illustrate a flow chart and pulse sequence for anexample of SWIFT.

FIG. 5 illustrates a flow chart to model introduction of artifactscaused by imperfections in an ideal pulse. FIG. 6 illustrates a timewiseview of excitation and acquisition using SWIFT.

Gapped pulse and gate effects can be viewed using simulated freeinduction decay (FID) data. The data indicates that gaps in thetransmitted pulse produce sidebands in the frequency domain excitationprofile. Gaps in the acquisition interval introduce sidebands andoverlap of the baseband in the received signal. Ripple on thetransmitter gate and receiver gate results in further intermodulationbetween the sidebands.

FIGS. 7A and 7B illustrate data having no gaps in the received data.FIG. 7A illustrates a received time domain signal in which there are notransmitter or receiver gaps. This view corresponds to “acq” from theSWIFT pulse sequence.

FIG. 7B illustrates a frequency domain signal (based on a Fouriertransform of the time domain signal).

FIGS. 8A and 8B illustrate data having gaps in the transmitted signalonly. FIG. 8A illustrates a time domain signal and includes highfrequency “ripples.” FIG. 8B illustrates a frequency domain signal. Theripples are due to the spin system's interaction with the transmittedpulse's sidebands. The interaction is only with the dispersion (Im)component of the spin system response.

FIGS. 9A and 9B illustrate gaps in both the transmitter side andreceiver side. FIG. 9A illustrates time domain data in which gaps arevisible in the received data. FIG. 9B illustrates a frequency domainsignal in which the tails of the sidebands overlap the baseband, thuscontributing to the bullseye artifact.

FIGS. 10A and 10B correspond to a simulated example. FIG. 10Aillustrates data for a system having ideal transmitter and receivergates. FIG. 10B depicts simulated ripple on the receive gate anddemonstrates that knowledge of a transmitted pulse is not sufficient tocorrect the data. Ripples in the data as shown in FIG. 10B may be causedby multiplicative effects.

FIG. 11 illustrates a top view of an object including a glass beakerfilled with water and a Teflon cube. The figure includes remnants of abullseye artifact.

FIGS. 12A and 12B illustrate the same object with bullseye correctionand inverse bullseye correction, as described herein.

FIGS. 13A and 13B illustrate a difference image of the same object. Theimage of FIG. 13A corresponds to subtraction of bullseye correction andbullseye correction followed by inverse bullseye correction whereas FIG.13B illustrates bullseye correction followed by inverse bullseyecorrection.

FIG. 14 illustrates a timewise plot comparing bullseye correction (lighttrace) and bullseye correction followed by inverse bullseye correction(dark trace). As shown, the plot corresponding to inverse bullseyecorrection has reduced ripple relative to the bullseye correction plot.

FIGS. 15A, 15B, and 15C illustrate sinograms corresponding to processingof the image data. A sinogram is a two-dimensional matrix that depictsconsecutive projections on the abscissa and frequency on the ordinate.The sinograms shown illustrate the effects of compensation and bullseyecorrection. The sinogram data can be transformed to recover an image.For example, the sinogram can be transformed to a k-space representationand by application of a multi-dimensional Fourier transform, convertedto an image.

FIG. 15A includes a sinogram representation in the frequency domain ofimage data for a radial projection. The data shown is uncompensated andnot corrected for a bullseye artifact. The uncompensated signalintensity can vary, for example, from zero to 50,000. With such a range,the effects of an outlying data point can undesirable mask otherintensity values and impair the image. In this view, the bullseyeartifact appears as alternating horizontal bands of dark and lightregions.

FIG. 15B includes a sinogram representation, also in the frequencydomain, and using a 1/t pre-compensation factor in the time domain. Asin the previous sinogram, the data shown is not corrected for a bullseyeartifact. For example, the signal intensity shown may have a range ofzero to 100, thus mitigating the effects of masking. In this view, thepersistent signal intensity modulation (bullseye artifact) appears asalternating horizontal bands of dark and light regions.

FIG. 15C includes a sinogram representation, also in the frequencydomain, having bullseye artifact correction. As in the previoussinogram, the data is pre-compensated using a 1/t factor. The absence ofalternating horizontal bands in the sinogram foretells the absence of abullseye artifact in the image.

The following portion describes gapped pulses for frequency-swept MRI.

Objects exhibiting extremely fast spin-spin relaxation rates can beimaged using a magnetic resonance (MR) method referred to as SWIFT(SWeep Imaging with Fourier Transform). SWIFT uses a frequency-sweptexcitation pulse and includes virtually simultaneous signal acquisitionin a time-shared mode. Correlation of the spin system response with theexcitation pulse function is used to extract the signals of interest.With SWIFT, image quality is related to uniform and broadband spinexcitation. A frequency-modulated pulse belonging to the hyperbolicsecant family (HSn pulses) can produce the excitation for SWIFT. Thisdocument describes implementation of SWIFT using HSn pulses and selectedproperties of HSn pulses in the rapid passage, linear region. Thisdocument also analyzes the pulses after inserting the “gaps” fortime-shared excitation and acquisition, presents an expression toestimate the amplitude and flip angle of the HSn pulses and the relativeenergy deposited by a SWIFT sequence.

SWIFT uses correlation of the spin system response with the excitationpulse function in order to extract useful signal.

Images can be produced with SWIFT provided that the excitation isuniform over a bandwidth equal to the image acquisition bandwidth.Certain types of frequency-modulated (FM) pulses that function accordingto adiabatic principles can produce a broadband and flat excitationprofile with low RF amplitude (B₁). Some FM pulses are suitable forinverting magnetization (i.e. adiabatic full passage (AFP)) and can alsobe used to excite lower flip angles while retaining essentially the sameshape of the frequency-response profile. In contrast to adiabaticinversion, excitation with lower flip angles entails either decreasingB₁ or increasing the rate at which the time-dependent pulse frequencyω_(RF)(t) is swept. In doing so, the operating point is changed from theadiabatic region to the region known as the rapid passage, linearregion, which satisfies the conditions:aT ₂ ²>>1  (6A)anda>>(ω₁/2π)²,  (6B)where a is the frequency acceleration in Hertz per second (i.e.,a=(dω_(RF)/dt)/2π), T₂ is spin-spin relaxation time in units of seconds,and ω₁ is the amplitude of RF field in angular frequency units (i.e.,ω₁=γB₁, where γ is the gyromagnetic ratio).

SWIFT can be implemented using a variety of shapes for the RF sweepfunction, and therefore, a variety of different kinds of FM pulses canbe used. For example, SWIFT can be implemented using FM pulses belongingto a class of hyperbolic secant pulses known as HSn pulses.

Frequency-modulated HSn pulses can be used for adiabatic inversion(i.e., AFP pulses) with reduced peak amplitude relative to a hyperbolicsecant (HS) pulse. The RF driving function ƒ_(n)(t) can be a modified HSfunction,ƒ_(n)(t)=sech(β(2t/T _(p)−1)^(n)),  (7)where n is a dimensionless shape factor (typically n≧1), β is adimensionless truncation factor (usually β≈5.3), and T_(p) is the pulselength (i.e., 0≦t≦T_(p)). The dimensionless relative integral I(n) andrelative power P(n) of the driving function (Equation 7) can beobtained, but does not have a known analytic closed form expressionswhen n>1. For convenience, here approximations will be used which forβ≧3 and n≧1 have 3% or better accuracy, such that:

$\begin{matrix}{{I(n)} = {{\frac{2}{T_{p}}{\int_{0}^{T_{p}}{{f_{n}(\tau)}{\mathbb{d}\tau}}}} \approx \left( \frac{\pi}{2\beta} \right)^{1/n}}} & \left( {8A} \right) \\{{P(n)} = {{\frac{2}{T_{p}}{\int_{0}^{T_{p}}{{f_{n}^{2}(\tau)}{\mathbb{d}\tau}}}} \approx {\left( \frac{1}{\beta} \right)^{1/n}.}}} & \left( {8B} \right)\end{matrix}$For n→∞ the function ƒ_(n)(t) becomes a rectangle which describes theshape of chirp and square pulses with corresponding parameters equal toone, i.e., I=P=1.

In the case of HSn pulses, the time-dependent RF amplitude and angularfrequency can be written as,

$\begin{matrix}{{{\omega_{1}(t)} = {\omega_{1\mspace{11mu}\max}{f_{n}(t)}}},} & \left( {9A} \right) \\{{{\omega_{RF}(t)} = {\omega_{c} + {2{A\left( {\frac{\int_{0}^{t}{{f_{n}^{2}(\tau)}{\mathbb{d}\tau}}}{\int_{0}^{T_{p}}{{f_{n}^{2}(\tau)}{\mathbb{d}\tau}}} - \frac{1}{2}} \right)}}}},} & \left( {9B} \right)\end{matrix}$respectively, where ω_(1 max)=γB_(1 max), ω_(c) is the angular carrierfrequency, and A is the amplitude of the frequency modulation. An NMRinstrument can execute FM pulses by modulating the pulse phase, ratherthan frequency, according to the function,

$\begin{matrix}{{\phi(t)} = {\int_{0}^{t}{\left( {{\omega_{RF}(\tau)} - \omega_{c}} \right){{\mathbb{d}\tau}.}}}} & (10)\end{matrix}$With amplitude and phase modulation, the FM pulse is described by thefunctionx(t)=ω₁(t)e ^(−jφ(t)).  (11)

By analyzing the vector motions in a rotating frame of reference, theexcitation bandwidth produced by an HSn pulse can be understood in termsof a frequency-swept excitation (−A≦(ω_(RF)(t)−ω_(c))≦A). Accordingly,the HSn pulse bandwidth (in Hz) is theoretically given by b_(w)=A/π.With uniform energy distribution inside the baseband, thefrequency-response profile of the HSn pulse is highly rectangular inshape, with edges becoming sharper with an increasing time-bandwidthproduct, R=AT_(p)/π.

Some features of HSn pulses can be determined by performing Blochsimulations.

FIG. 16 illustrates dependency of excitation bandwidth on the flip anglefor selected pulses. For HS1 and chirp pulses b_(w,theory)=A/π and forthe square pulse b_(w,theory)=1/T_(p). The bandwidth dependence isdisplayed as the ratio between the Bloch simulations calculatedbandwidth needed to achieving 95% maximal excitation and the theoreticalvalues from linear systems considerations.

FIG. 16 shows simulated data using HS1 and chirp pulses (R=256), and forcomparison, the data obtained with a simple square pulse are also shown.Bloch simulations can be used to find the bandwidth b_(w,95) for whichat least 95% of the maximal excitation was achieved. In FIG. 16,b_(w,95) is plotted as a function of flip angle θ, after normalizing byb_(w,theory), which is defined as the full-width half-maximum bandwidthpredicted from linear theory. In other words, b_(w,theory)=A/π for HSnpulses and b_(w,theory)=1/T_(p) for a square pulse. In FIG. 16, it canbe seen that the HS1 and chirp bandwidths are well approximated by therelationship b_(w)≈A/π=R/T_(p), for all flip angles (i.e.,b_(w,95)/b_(w,theory)≈1). In comparison, the excitation bandwidthproduced by a square pulse is highly dependent on the choice of flipangle. In the small flip angle region, the bandwidth of the squarepulse, as predicted from Fourier analysis under estimates b_(w,95) byabout a factor of three. Thus, to compare with HSn pulses, the squarepulse used in the following analysis will have a pulse lengthT_(p)=1/(3b_(w)), so that it effectively excites the same requiredbandwidth (b_(w,95)) as the HSn pulse.

In the linear region, the flip angle produced by an HSn pulse is alinear function of the RF field strength. The RF field strengthexpressed in terms of the spectral density at the center frequency (j₀)is

$\begin{matrix}{j_{0} = {{{\int_{0}^{T_{p}}{{x(t)}{\mathbb{d}t}}}}.}} & (12)\end{matrix}$When using amplitude-modulated (AM) pulses without frequency or phasemodulation (e.g., a sinc pulse), j₀ is a linear function of T_(p)according to Equation 12. Alternatively, with frequency swept pulses j₀can be a non-linear function of T_(p), due to the flexibility providedby the additional parameter, R. For example, with a chirp pulse j₀ isinversely proportional to the square root of the frequency acceleration:j ₀∝1/√{square root over (a)}=√{square root over ((πT _(p))/A)}≈√{squareroot over (T _(p) /b _(w))}.  (13)

As members of the same FM pulse family, HSn pulses are expected toexhibit similar j₀-dependency on a, although with slight differences dueto their altered shapes.

FIG. 17 illustrates flip angles simulated for five HSn pulses withdifferent parameters in the range of changing ω_(1 max) values from300-39000 rad/s. The different symbols represent different set ofparameters n, β R, and T_(p), which are respectively: 1, 7.6, 256, 3 ms(square), 1, 2.99, 256, 3 ms (circle), 1, 5.3 256, 1 ms (up triangle),1, 5.3 256, 3 ms (down triangle), 8, 5.3, 256, 1 ms (diamond), and chirpwith R=256 and T_(p)=2.56 ms (cross). The line represents Equations14A-14C.

As shown in FIG. 17, results from Bloch simulations demonstrate how HSnpulses follow these expectations. In FIG. 17, the simulated flip angles(θ) obtained with different settings of the pulse parameters (n, β,T_(p), and b_(w)) are plotted on the ordinate, while the predicted flipangles (θ′) based on the analytical approximation,

$\begin{matrix}{{\theta^{\prime} = {{\omega_{1\mspace{14mu}\max}\beta^{{{- 1}/2}n}\sqrt{\frac{T_{p}}{b_{w}}}} \approx j_{0}}},} & \left( {14A} \right)\end{matrix}$are plotted on the abscissa. Here β^(−1/2n), describes the shape factor,which according to Equation 8, is related to both the relative integraland power of the pulse as

$\beta^{{{- 1}/2}n} \approx {\left( \frac{\pi}{2} \right)^{{{- 1}/2}n}\sqrt{I(n)}} \approx \sqrt{P(n)}$and is equal to 1 in the case of chirp. The approximation θ≈θ′ holds forflip angles up to π/2 with an accuracy of about 3%.

Alternative equations to approximate the flip angle are

$\begin{matrix}{{\theta^{\prime} = {{\omega_{1\max}\beta^{{{- 1}/2}n}\frac{T_{p}}{\sqrt{R}}} \approx \theta}}{and}} & \left( {14B} \right) \\{\theta^{\prime} = {{\omega_{1\max}\beta^{{{- 1}/2}n}\frac{\sqrt{R}}{b_{w}}} \approx {\theta.}}} & \left( {14C} \right)\end{matrix}$Based on the application, a different choice of dependent versusindependent parameters can be made, thus leading to using Equations 14A,14B, or 14C. Some MRI systems implement pulses as a “shape file” havinga fixed R value with the constraint of Equations 14B or 14C.

For comparison with the HSn pulses, consider a square pulse havingapproximately the same excitation bandwidth b_(w). As described above,when requiring the magnitude of excited magnetization at the edges ofthe frequency-response profile to be at least 95% of maximum, the pulselength in the linear region is about

$\frac{1}{3b_{w}}$(see FIG. 16). The flip angle of such a square pulse with peak amplitudeω_(1max) is equal to

$\begin{matrix}{\theta_{square} = {\frac{\omega_{1\;\max}}{3b_{w}}.}} & (15)\end{matrix}$The peak amplitudes for excitation to a flip angle θ using HSn andsquare pulses are

$\begin{matrix}{{\omega_{1\max}^{HSn} \approx {{\theta\beta}^{{1/2}n}\sqrt{\frac{b_{w}}{T_{p}}}}} = {{\theta\beta}^{{1/2}n}\frac{b_{w}}{\sqrt{R}}}} & (16)\end{matrix}$andω_(1max) ^(square)≈3b _(w)θ,  (17)respectively. In contrast to the square pulse, HSn pulses can producethe same θ and b_(w) values with different settings of the peak RFamplitude, ω_(1max) ^(HSn). The relative peak amplitude required by thesquare and HSn pulses is given by the ratio,

$\begin{matrix}{{\frac{\omega_{1\max}^{square}}{\omega_{1\max}^{HSn}} \approx {3\beta^{{{- 1}/2}n}\sqrt{R}}},} & (18)\end{matrix}$which can reach large values, depending on the R value. For example, theHS8 pulse with R=512 and β=5.3 has 61 times less peak amplitude than asquare pulse exciting the same bandwidth at the same flip angle.

The relative energy, J, radiated by any RF pulse is proportional to thepower and duration of the pulse. For an HSn pulse the energy is:

$\begin{matrix}{J_{HSn} = {{{\left( \omega_{1\max} \right)^{2}{P(n)}T_{p}} \approx {\left( {\beta^{{1/2}n}\theta\sqrt{\frac{b_{w}}{T_{p}}}} \right)^{2}\frac{T_{p}}{\sqrt[n]{\beta}}}} = {\theta^{2}b_{w}}}} & (19)\end{matrix}$and accordingly for a square pulse:J _(square)=(3b _(w)θ)² T _(p)=3θ² b _(w).  (20)Thus, the radiated RF energy of an HSn pulse is not dependent on n, peakamplitude, or pulse length, and is at least 3 times less than the energyradiated by a square pulse exciting the same bandwidth at the same flipangle.

To generate a shaped frequency-modulated pulse, an NMR spectrometer canuse a discrete representation of the pulse with a finite number of pulseelements. Consider next the total number of pulse elements (N_(tot)) forproper representation of the pulse. Mathematically such pulse, x′(t),can be represented as the multiplication of the continuous RF pulsefunction, x(t), by a comb function of spacing Δt (Δt=T_(p)/N_(tot)), andconvolving the result with a rectangle function having the same widthΔt:x′(t)=[x(t)comb(t/Δt)]⊕rect(t/Δt),  (21)where ⊕ is the convolution operation. The Fourier Transform of x′(t)represents the “low flip angle” excitation profile of the discretizedpulse in a frequency (v) domain,X′(v)=B[X(v)⊕comb(vΔt)]sinc (vΔt),  (22)where B=Δt² is a scaling factor which may be neglected below forsimplicity.

Thus, discretization creates an infinite number of sidebands having thesame bandwidth b_(w), centered at frequencies k/Δt where k is aninteger. The entire excitation spectrum is weighted by the envelopefunction, sinc

$\left( {v\;\Delta\; t} \right) = {\frac{\sin\left( {v\;\pi\;\Delta\; t} \right)}{v\;\pi\;\Delta\; t}.}$

The Nyquist condition for a discretized excitation determines that thesidebands are not aliased when1/Δt≧b _(w).  (23)

This in turn determines the minimum number of pulse elements to satisfythe Nyquist condition, N_(Nyquist), which depends on R according to:

$\begin{matrix}{N_{Nyquist} = {\frac{T_{p}}{\Delta\; t} = {\frac{R}{b_{w}\Delta\; t} = {R.}}}} & (24)\end{matrix}$

If the number of pulse elements satisfies the Nyquist condition, thenthe baseband (−b_(w)/2≦v≦b_(w)/2) can be described byX′ ^(main)(v)=X(v) sinc (vΔt).  (25)To reduce the attenuation by sinc (vΔt), the length of Δt can bedecreased (i.e. the pulse can be oversampled). The parameter used tocharacterize the level of pulse oversampling is

$\begin{matrix}{L_{over} = {\frac{N_{tot}}{N_{Nyquist}} = {\frac{N_{tot}}{R}.}}} & (26)\end{matrix}$To meet the requirement that the pulse has at least 95% maximumexcitation at the edges of the baseband, L_(over) must be at least 3,which is similar in form to the constraint in square pulse excitation.One example uses an even larger oversampling level.

FIGS. 18A-18J graphical present the components of Equation 24 (18A-18C,18F-18H) and Equation 29 (18D, 18E, 18I, 18J) for two different pulseoversampling levels L_(over)=1 (18A-18E) and L_(over)=8 (18F-18J). Inthe case of the gapped pulse (18D, 18E, 18I, 18J), the duty cycle, d_(c)is equal to 0.5.

The different components of Equation 22 are presented graphically inFIGS. 18A-18J for two different pulse oversampling levels, L_(over)=1(FIGS. 18A-18C) and L_(over)=8 (FIGS. 18F-18H). With L_(over)=1 theresulting excitation spectrum has its first sidebands locatedimmediately adjacent to the baseband (FIG. 18C), and for L_(over)=8 thefirst sidebands are pushed outward to center on frequencies ±8b_(w)(FIG. 18H).

In SWIFT, the transmitter is repeatedly turned on and off (every dwinterval) to enable sampling “during” the pulse. Such full amplitudemodulation of the excitation pulse creates modulation sidebands worthyof consideration. If the “transmitter on” time is labeled as τ_(p), thenthe time with the “transmitter off” is equal to dw−τ_(p). The pulse isdivided into a number of segments (N_(seg)) each of duration dw, and thetotal number of sampling points (N_(samp)) is a multiple of N_(seg):N _(samp) =N _(seg) S _(over),  (27)where S_(over) is the integer describing the acquisition oversampling.In general, the parameter N_(seg) is not dependent on N_(tot) with therestriction, N_(tot)≧2N_(seg).

FIGS. 19A-19C illustrate shaped pulses with gaps for acquisition in theSWIFT sequence (FIG. 19A) and detailed structure of the pulse withdifferent pulse oversampling levels, L_(over)=2 (FIG. 19B) andL_(over)=6 (FIG. 19C). In both examples, the pulses have a duty cycle

$d_{c} = \frac{\tau_{p}}{dw}$equal to 0.5. The timing of the gapped HSn pulse as used in SWIFT ispresented in FIG. 19A.

There are different ways to create such segmented pulses based on thesame parent pulse (Equation 11). One way is to introduce delays withzero amplitudes into the parent pulse (DANTE type) and another involveszeroing pulse elements in the parent pulse (gapped pulse). According toBloch simulations, the excitation profile of pulses created in these twodifferent ways are the same for R≦N_(seg)/2. For the same pulse lengthand duty cycle, the gapped pulse shows better behavior (flatterexcitation profile) up to the maximum usable R value, which is R=N_(seg)(Equation 24). Consider, for example, gapped pulses.

Mathematically the gapped pulse can be described as:x′ _(gap)(t)=x′(t)(comb(t/dw)⊕rect(t/τ _(p))).  (28)

The Fourier Transform of x′_(gap)(t) represents the “low flip angle”excitation profile of the gapped pulse:X′ _(gap)(v)=X′(v)⊕[comb(vdw) sinc (vτ _(p))].  (29)

The different components of this equation are presented graphically inFIGS. 18A-18J for two different levels of pulse oversampling, L_(over).Decreasing Δt pushes the sidebands farther from the baseband, butconvolution with function comb(vdw) sinc (vτ_(p)) brings the sidebandsback with amplitudes weighted by sinc (m d_(c)), where m is the sidebandorder. As a result of this convolution, the profile of the baseband ischanged by sideband contamination.

FIGS. 20A and 20B show profiles of the baseband produced by gappedpulses with two different oversampling levels. Sideband contaminationimpairs the flatness of the profile, especially at the edges of thebaseband. Increasing the level of pulse oversampling helps to decreaseor eliminate this effect. This effect becomes negligible withL_(over)≧16, at which point the baseband and sideband profiles becomeflat with amplitudes equal to:A _(m)=sinc (md _(c)).  (30)As the duty cycle decreases (d_(c)→0), the amplitude of sidebandsapproaches the amplitude of the baseband, whereas the sidebandsdisappear as d_(c)→1.

Inserting gaps in an HSn pulse does not change the baseband excitationbandwidth and decreases the flip angle proportionally to the duty cycle.To make the same flip angle, the peak amplitude of the pulse isincreased, and the resulting energy of the pulse increases by

$\frac{1}{d_{c}}.$Formulas accounting for the duty cycle are given in Table 2.

TABLE 2 Properties of gapped HSn and square pulses Gapped HSn pulsesParameter (d_(c)-duty cycle) Square pulse RF driving function, f_(n)(t)sech (β(2t/T_(p) − 1)^(n)) Constant Relative RF integral, I(n)${\frac{2\; d_{c}}{T_{p}}{\int_{0}^{T_{p}}{{f_{n}(\tau)}\ {d\tau}}}} \approx {d_{c}\left( \frac{\pi}{2\;\beta} \right)}^{1/n}$1 n → ∞ Relative RF power, P(n)${\frac{2\; d_{c}}{T_{p}}{\int_{0}^{T_{p}}{{f_{n}^{2}(\tau)}\ {d\tau}}}} \approx {d_{c}\left( \frac{1}{\beta} \right)}^{1/n}$1 n → ∞ Total pulse length, T_(p), s $\frac{R}{b_{w}}$$\frac{1}{3\; b_{w}}$ Amplitude modulation ω_(1max)f_(n)(t) ω_(1max)rectfunction, ω₁(t) [comb(b_(w)t)⊕rect(b_(w)d_(c)t)] (t/T_(p)) Frequencymodulation function, ω_(RF)(t), rad/s$\omega_{c} + {2\;{A\left( {\frac{\int_{0}^{t}{{f_{n}^{2}(\tau)}d\;\tau}}{\int_{0}^{T_{p}}{{f_{n}^{2}(\tau)}\mspace{11mu} d\;\tau}} - \frac{1}{2}} \right)}}$Constant Flip angle, θ, rad$\approx {\omega_{1\;\max}d_{c}\beta^{{{- 1}/2}\; n}\frac{\sqrt{R}}{b_{w}}}$$\approx \frac{\omega_{1\;\max}}{3\; b_{w}}$ Peak RF amplitude,ω_(1max), rad/s$\approx {\frac{\beta^{{1/2}\; n}}{d_{c}\sqrt{R}}\theta\; b_{w}}$≈3θb_(w) Relative RF energy, J, rad²/s$\approx {\frac{1}{d_{c}}\theta^{2}b_{w}}$ ≈3θ²b_(w) Relative RF energy(Ernst angle), J, rad²/s $\approx \frac{2\; T_{R}b_{w}}{T_{1}d_{c}}$$\approx \frac{6\; T_{R}b_{w}}{T_{1}}$ Relative SAR (Ernst angle), SAR,rad²/s² $\approx \frac{2\; b_{w}}{T_{1}d_{c}}$$\approx \frac{6\; b_{w}}{T_{1}}$ Relative amplitudes of ≈sinc(md_(c)) —sidebands, A_(m)

Consider the RF energy deposition during a particular SWIFT sequence.The maximum signal/noise (S/N) ratio is reached when the flip angle isadjusted to the “Ernst angle” (θ_(E)), which is equal toθ_(E)=arccos(e^(−T) ^(R) ^(/T) ^(l) ), where T_(R) is repetition timeand T_(l) is longitudinal relaxation time. An approximation for theErnst angle in rapid NMR sequences (T_(R)/T_(l)<0.1) is:

$\begin{matrix}{\theta_{E} \approx {\sqrt{\frac{2T_{R}}{T_{1}}}.}} & (31)\end{matrix}$In this case, the relative energy deposition according to Equation 19 isequal to:

$\begin{matrix}{{J_{HSn} \approx \frac{2T_{R}b_{w}}{T_{1}d_{c}}},} & (32)\end{matrix}$and the relative SAR for the HSn pulses is:

$\begin{matrix}{{{SAR}_{HSn} \approx \frac{2b_{w}}{T_{1}d_{c}}},} & (33)\end{matrix}$and for a square pulse is:

$\begin{matrix}{{SAR}_{square} \approx {\frac{6b_{w}}{T_{1}}.}} & (34)\end{matrix}$

Thus, the energy and SAR of a gapped HSn pulse at the Ernst angle isindependent of pulse length, R value, and the specific pulse shape (n),and is linearly proportional to pulse baseband width. As compared with asquare pulse producing the same bandwidth (b_(w,95)), HSn pulses havereduced SAR when the duty cycle (d_(c)) is large and greater SAR whend_(c)<0.33.

Note that in general, with decreasing pulse duty cycle, the S/Nincreases together with the SAR, which under certain circumstances leadsto a compromise in the choice of “optimum” duty cycle.

In SWIFT, useful signal is extracted by correlation of the spin systemresponse with the pulse function. A misrepresentation of the pulsefunction will result in a systematic error in the resulting spectrum.Such a systematic error in radial imaging shows up in the reconstructedimage as a “bullseye” artifact. To avoid such artifacts, the pulsefunction used for correlation should be as faithful as possible to thephysical pulse transmitted by the RF coil. Some errors can be predictedand neutralized on a software level:

-   -   1. Gapping effects. For correlation, instead of the theoretical        function, x(t), the discretized function x′_(gap)(t), is used.    -   2. Digitization. The function, x′_(gap)(t), can be rounded by        software and hardware.    -   3. Timing errors. To avoid temporal rounding error the pulse        element duration Δt can be equal to or be a multiple of the        minimum time step (temporal resolution) of the waveform        generator.

Additional Notes

Example 1 includes a method. The method includes receiving andcalculating. The method includes receiving image data. The image dataincludes a radial sampled projection of an object. The method includescalculating, using a processor, a first corrected data set using theimage data and using a first ratio. The first ratio is based on anaverage projection of the image data and is based on a filtered averageprojection of the image data. The method includes calculating a secondcorrected data set using the first corrected data set and using a secondratio. The second ratio is based on an average projection of the firstcorrected data set and a filtered average projection of the firstcorrected data set.

Example 2 includes Example 1 wherein calculating the second correcteddata set includes using an inverse of the second ratio.

Example 3 includes any one of Examples 1 or 2 wherein calculating thesecond corrected data set includes multiplying using a weighting factor.

Example 4 includes any one of Examples 1 to 3 further includingrepeating at least one of calculating the first corrected data set orcalculating the second corrected data set.

Example 5 includes any one of Examples 1 to 4 further includingiteratively calculating the first corrected data set and calculating thesecond corrected data set.

Example 6 includes any one of Examples 1 to 5 wherein determining thefiltered average projection includes using a low pass filter.

Example 7 includes any one of Examples 1 to 6 wherein receiving imagedata includes receiving magnetic resonance data generated using SWIFTprotocol.

Example 8 includes any one of Examples 1 to 7 further including usingthe second corrected data set to generate a corrected image.

Example 9 includes any one of Examples 1 to 8 further including printingthe corrected image using a printer or displaying the corrected imageusing a display.

Example 10 includes a system comprising a memory and a processor. Thememory is configured to receive image data corresponding to a radialprojection of an object. The processor is coupled to the memory. Theprocessor is configured to calculate a first corrected data set andcalculate a second corrected data. The first corrected data set iscalculated using the image data and using a first ratio. The first ratiois based on an average projection of the image data and based on afiltered average projection of the image data. The second corrected dataset is calculated using the first corrected data set and using a secondratio. The second ratio is based on an average projection of the firstcorrected data set and a filtered average projection of the firstcorrected data set.

Example 11 includes Example 10 wherein the processor is configured tocalculate at least one of the first corrected data set or the secondcorrected data set using at least one of multiplication in a frequencydomain or convolution in a time domain.

Example 12 includes any one of Examples 10 or 11 and further including adisplay coupled to the processor and configured to render a visibleimage of the object using the corrected data.

Example 13 includes any one of Examples 10 to 12 wherein the processoris configured to execute an algorithm to estimate a persistentdistortion in an excitation signal provided to the object.

Example 14 includes any one of Examples 10 to 14 wherein the processoris configured to implement a low pass filter.

Example 15 includes a machine-readable medium having machine executableinstructions stored thereon that causes one or more processors toperform operations including receiving and calculating. The medium hasinstructions for receiving image data including a radial sampledprojection of an object. The medium has instructions for calculating,using a processor, a first corrected data set using the image data andusing a first ratio. The first ratio is based on an average projectionof the image data and based on a filtered average projection of theimage data. The medium has instructions for calculating a secondcorrected data set using the first corrected data set and using a secondratio. The second ratio is based on an average projection of the firstcorrected data set and a filtered average projection of the firstcorrected data set.

Example 16 includes Example 15 wherein receiving the image data includesreceiving magnetic resonance data corresponding to an adiabatic pulse.

Example 17 includes any one of Examples 15 or 16 wherein determining thefiltered average projection includes estimating.

Example 18 includes any one of Examples 15 to 17 wherein determining thefiltered average projection includes implementing a low pass filter.

Example 19 includes any one of Examples 15 to 18 further includinggenerating a corrected image based on the second corrected data set.

Example 20 includes any one of Examples 15 to 19 wherein receiving theimage data includes receiving magnetic resonance data generated usingSWIFT protocol.

The above detailed description includes references to the accompanyingdrawings, which form a part of the detailed description. The drawingsshow, by way of illustration, specific embodiments in which theinvention can be practiced. These embodiments are also referred toherein as “examples.” Such examples can include elements in addition tothose shown and described. However, the present inventors alsocontemplate examples in which only those elements shown and describedare provided.

All publications, patents, and patent documents referred to in thisdocument are incorporated by reference herein in their entirety, asthough individually incorporated by reference. In the event ofinconsistent usages between this document and those documents soincorporated by reference, the usage in the incorporated reference(s)should be considered supplementary to that of this document; forirreconcilable inconsistencies, the usage in this document controls.

In this document, the terms “a” or “an” are used, as is common in patentdocuments, to include one or more than one, independent of any otherinstances or usages of “at least one” or “one or more.” In thisdocument, the term “or” is used to refer to a nonexclusive or, such that“A or B” includes “A but not B,” “B but not A,” and “A and B,” unlessotherwise indicated. In the appended claims, the terms “including” and“in which” are used as the plain-English equivalents of the respectiveterms “comprising” and “wherein.” Also, in the following claims, theterms “including” and “comprising” are open-ended, that is, a system,device, article, or process that includes elements in addition to thoselisted after such a term in a claim are still deemed to fall within thescope of that claim. Moreover, in the following claims, the terms“first,” “second,” and “third,” etc. are used merely as labels, and arenot intended to impose numerical requirements on their objects.

Method examples described herein can be machine or computer-implementedat least in part. Some examples can include a computer-readable mediumor machine-readable medium encoded with instructions operable toconfigure an electronic device to perform methods as described in theabove examples. An implementation of such methods can include code, suchas microcode, assembly language code, a higher-level language code, orthe like. Such code can include computer readable instructions forperforming various methods. The code may form portions of computerprogram products. Further, the code may be tangibly stored on one ormore volatile or non-volatile computer-readable media during executionor at other times. These computer-readable media may include, but arenot limited to, hard disks, removable magnetic disks, removable opticaldisks (e.g., compact disks and digital video disks), magnetic cassettes,memory cards or sticks, random access memories (RAMs), read onlymemories (ROMs), and the like.

The above description is intended to be illustrative, and notrestrictive. For example, the above-described examples (or one or moreaspects thereof) may be used in combination with each other. Otherembodiments can be used, such as by one of ordinary skill in the artupon reviewing the above description. The Abstract is provided to complywith 37 C.F.R. §1.72(b), to allow the reader to quickly ascertain thenature of the technical disclosure. It is submitted with theunderstanding that it will not be used to interpret or limit the scopeor meaning of the claims. Also, in the above Detailed Description,various features may be grouped together to streamline the disclosure.This should not be interpreted as intending that an unclaimed disclosedfeature is essential to any claim. Rather, inventive subject matter maylie in less than all features of a particular disclosed embodiment.Thus, the following claims are hereby incorporated into the DetailedDescription, with each claim standing on its own as a separateembodiment. The scope of the invention should be determined withreference to the appended claims, along with the full scope ofequivalents to which such claims are entitled.

What is claimed is:
 1. A method comprising: receiving image dataincluding a radial sampled projection of an object; calculating, using aprocessor, a first corrected data set using the image data and using afirst ratio, the first ratio based on an average projection of the imagedata and based on a filtered average projection of the image data; andcalculating a second corrected data set using the first corrected dataset and using an inverse of a second ratio, the second ratio based on anaverage projection of the first corrected data set and a filteredaverage projection of the first corrected data set; and using the secondcorrected data set to generate a corrected image.
 2. The method of claim1 wherein calculating the second corrected data set includes multiplyingusing a weighting factor.
 3. The method of claim 1 further includingrepeating at least one of calculating the first corrected data set orcalculating the second corrected data set.
 4. The method of claim 1further including iteratively calculating the first corrected data setand calculating the second corrected data set.
 5. The method of claim 1wherein determining the filtered average projection of the image data ordetermining the filtered average projection of the first corrected dataset includes using a low pass filter.
 6. The method of claim 1 whereinreceiving image data includes receiving magnetic resonance datagenerated using SWIFT protocol.
 7. The method of claim 1 furtherincluding printing the corrected image using a printer or displaying thecorrected image using a display.
 8. A system comprising: a memoryconfigured to receive image data corresponding to a radial projection ofan object; and a processor coupled to the memory, the processorconfigured to calculate a first corrected data set and calculate asecond corrected data, wherein the first corrected data set iscalculated using the image data and using a first ratio, the first ratiobased on an average projection of the image data and based on a filteredaverage projection of the image data and wherein the second correcteddata set is calculated using the first corrected data set and using aninverse of a second ratio, the second ratio based on an averageprojection of the first corrected data set and a filtered averageprojection of the first corrected data set, and the second correcteddata set is used to generate a corrected image.
 9. The system of claim 8wherein the processor is configured to calculate at least one of thefirst corrected data set or the second corrected data set using at leastone of multiplication in a frequency domain or convolution in a timedomain.
 10. The system of claim 8 further including a display coupled tothe processor and configured to render a visible image of the objectusing the second corrected data set.
 11. The system of claim 8 whereinthe processor is configured to execute an algorithm to estimate apersistent distortion in an excitation signal provided to the object.12. The system of claim 8 wherein the processor is configured toimplement a low pass filter.
 13. A non-transitory machine-readablemedium having machine executable instructions stored thereon that causesone or more processors to perform operations including: receiving imagedata including a radial sampled projection of an object; calculating,using a processor, a first corrected data set using the image data andusing a first ratio, the first ratio based on an average projection ofthe image data and based on a filtered average projection of the imagedata; and calculating a second corrected data set using the firstcorrected data set and using an inverse of a second ratio, the secondratio based on an average projection of the first corrected data set anda filtered average projection of the first corrected data set; andgenerating a corrected image based on the second corrected data set. 14.The machine-readable medium of claim 13 wherein receiving the image dataincludes receiving magnetic resonance data corresponding to an adiabaticpulse.
 15. The machine-readable medium of claim 13 wherein determiningthe filtered average projection of the image data or determining thefiltered average projection of the first corrected data set includesestimating.
 16. The machine-readable medium of claim 13 whereindetermining the filtered average projection of the image data ordetermining the filtered average projection of the first corrected dataset includes implementing a low pass filter.
 17. The machine-readablemedium of claim 13 wherein receiving the image data includes receivingmagnetic resonance data generated using SWIFT protocol.